Mathematical Air Pollution Models: Eulerian Models
139
Nieuwstadt (1980) presented a solution, which was a particular case of Smith’s
(1975b) solution noted above. Subsequently, Nieuwstadt and de Haan (1981) extended
that solution to the case of a growing boundary layer height. Catalano (1982), in turn,
extended the latter solution to the case of nonzero mean vertical wind profi les. Lin
and Hildemann (1997) extended the solution of Demuth (1978) with boundary conditions suitable for simulating dry deposition to the ground.
Recently, Brown et al. (1997) derived equations for point source releases for the
fi rst four moments of the vertical concentration distribution and the magnitude and
downwind location of the maximum ground concentration from the solution of Yeh
and Huang (1975).
Finally, Moreira et al. (2005b) found a general two-dimensional steady-state solution for any profi les of wind and eddy coeffi cient diffusions.
5.3.3 NUMERICAL SOLUTIONS
Among the techniques used to resolve Equation 5.5, the following should be
mentioned:
Finite difference method
•
Finite elements method
•
Finite volume method
•
Spectral methods
•
The method of confi ned elements
•
The fi nite difference method is the most simple technique and was the fi rst to be
used. The approximation of the fi nite differences of the advection term
–
u ∙ ∇
– c , is,
however, always associated with an error that artifi cially increases diffusion in the
fi nal results of the simulated concentrations. Several techniques have been developed
with the aim of reducing this error, and notwithstanding the limitations posed by
this model, it remains one of the most important and widely adopted methods of
simulation.
Unlike analytic approaches, numerical techniques allow, from the theoretical
viewpoint, the use of any function for K(x, y, z, t).
5.3.4 EDDY DIFFUSIVITY
The literature proposes several expressions for K z , which is generally a function of
height z (for instance, Pleim and Chang, 1992). For instance, an approach for estimating the eddy diffusivity K and dispersion parameters as functions of eddy scale size
in the PBL and relative amount of turbulent energy has been recently proposed by
Degrazia and Moraes (1992) and Degrazia et al. (1997, 2000). Making use of Taylor’s
statistical theory (Taylor, 1921), the Hay and Pasquill working approximation of the
relationship between Lagrangian and Eulerian turbulence spectra (Hay and Pasquill,
1959), and a model for Eulerian spectra, such approach relates plume dispersion in a
boundary layer mainly to the turbulent eddies acting in the different stability regimes
of the boundary layer (Pasquill and Smith, 1983). Bearing the K-theory limitations in
© 2010 by Taylor and Francis Group, LLC
139
Nieuwstadt (1980) presented a solution, which was a particular case of Smith’s
(1975b) solution noted above. Subsequently, Nieuwstadt and de Haan (1981) extended
that solution to the case of a growing boundary layer height. Catalano (1982), in turn,
extended the latter solution to the case of nonzero mean vertical wind profi les. Lin
and Hildemann (1997) extended the solution of Demuth (1978) with boundary conditions suitable for simulating dry deposition to the ground.
Recently, Brown et al. (1997) derived equations for point source releases for the
fi rst four moments of the vertical concentration distribution and the magnitude and
downwind location of the maximum ground concentration from the solution of Yeh
and Huang (1975).
Finally, Moreira et al. (2005b) found a general two-dimensional steady-state solution for any profi les of wind and eddy coeffi cient diffusions.
5.3.3 NUMERICAL SOLUTIONS
Among the techniques used to resolve Equation 5.5, the following should be
mentioned:
Finite difference method
•
Finite elements method
•
Finite volume method
•
Spectral methods
•
The method of confi ned elements
•
The fi nite difference method is the most simple technique and was the fi rst to be
used. The approximation of the fi nite differences of the advection term
–
u ∙ ∇
– c , is,
however, always associated with an error that artifi cially increases diffusion in the
fi nal results of the simulated concentrations. Several techniques have been developed
with the aim of reducing this error, and notwithstanding the limitations posed by
this model, it remains one of the most important and widely adopted methods of
simulation.
Unlike analytic approaches, numerical techniques allow, from the theoretical
viewpoint, the use of any function for K(x, y, z, t).
5.3.4 EDDY DIFFUSIVITY
The literature proposes several expressions for K z , which is generally a function of
height z (for instance, Pleim and Chang, 1992). For instance, an approach for estimating the eddy diffusivity K and dispersion parameters as functions of eddy scale size
in the PBL and relative amount of turbulent energy has been recently proposed by
Degrazia and Moraes (1992) and Degrazia et al. (1997, 2000). Making use of Taylor’s
statistical theory (Taylor, 1921), the Hay and Pasquill working approximation of the
relationship between Lagrangian and Eulerian turbulence spectra (Hay and Pasquill,
1959), and a model for Eulerian spectra, such approach relates plume dispersion in a
boundary layer mainly to the turbulent eddies acting in the different stability regimes
of the boundary layer (Pasquill and Smith, 1983). Bearing the K-theory limitations in
© 2010 by Taylor and Francis Group, LLC